Points, Lines, and Angles: Additional Practice (Set 2)

Vertical angles are the opposite angles formed when two lines cross.

They are always equal in measure; they need not be supplementary or complementary, and they do not form a linear pair.

Complementary angles sum to $90^\circ$. Let the complement be $x$, so the angle is $x + 24$.

$x + (x + 24) = 90 \implies 2x = 66 \implies x = 33$

The angle is $33^\circ + 24^\circ = 57^\circ$ (its complement is $33^\circ$).

Complementary angles sum to $90^\circ$. Write the angles as $k$ and $4k$.

$k + 4k = 90^\circ \implies 5k = 90^\circ \implies k = 18^\circ$

The larger angle is $4(18) = 72^\circ$.

Supplementary angles sum to $180^\circ$. If the angle equals its supplement, call each $x$.

$x + x = 180 \implies 2x = 180 \implies x = 90$

A right angle is the only angle equal to its own supplement.

A linear pair is supplementary. Let the smaller be $x$, so the larger is $5x$.

$x + 5x = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ$

The smaller angle is $30^\circ$ (the larger is $150^\circ$).

The angle is $x$ and its supplement is $180^\circ - x$. The supplement is $4$ times the angle.

$180 - x = 4x \implies 180 = 5x \implies x = 36$

The angle is $36^\circ$; its supplement is $144^\circ$, which is indeed $4 \times 36^\circ$.

Co-interior (same-side interior) angles between parallel lines are supplementary, so they sum to $180^\circ$.

$2x + 3x = 180^\circ \implies 5x = 180^\circ \implies x = 36^\circ$

The larger angle is $3(36) = 108^\circ$.

The angle is $x$, its complement is $90 - x$, and its supplement is $180 - x$. The complement is one-fourth of the supplement.

$90 - x = \tfrac{1}{4}(180 - x) \implies 360 - 4x = 180 - x \implies 3x = 180 \implies x = 60$

Check: the complement is $30^\circ$ and the supplement is $120^\circ$, and $30^\circ$ is one-fourth of $120^\circ$.

When parallel lines are cut by a transversal, corresponding angles are equal.

$4x = x + 60 \implies 3x = 60 \implies x = 20$

Check: each angle is $4(20) = 80^\circ$, matching $(20) + 60 = 80^\circ$.

Supplementary angles sum to $180^\circ$. Let the smaller be $x$; the larger is $x + 90$.

$x + (x + 90) = 180 \implies 2x = 90 \implies x = 45$

The larger angle is $45^\circ + 90^\circ = 135^\circ$ (the smaller is $45^\circ$).

Vertical angles are equal, so angle $B = 130^\circ$. A linear pair is supplementary, so $C = 180^\circ - 130^\circ = 50^\circ$.

$B + C = 130^\circ + 50^\circ = 180^\circ$

(Angles $B$ and $C$ are themselves a linear pair, which is why the sum is $180^\circ$.)

Angles around a point sum to $360^\circ$.

$x + 2x + 4x + 3x = 360 \implies 10x = 360 \implies x = 36$

The largest angle is $4(36) = 144^\circ$.